In triangle ABC, the measure of ∠B is 90°,
BC = 16, and AC = 20. Triangle DEF is similar to triangle ABC, where vertices D, E, and F correspond to vertices A, B, and C, respectively, and each side of triangle DEF is 1/3 the length of the corresponding side of triangle ABC. What is the value of sinF?
Since ∠B is a right angle, we have sin(B) = 1 and cos(B) = 0.
Using the Pythagorean theorem, we can find the length of the third side of triangle ABC:
AB^2 = AC^2 - BC^2 = 20^2 - 16^2 = 144
So AB = 12.
Now, since triangles DEF and ABC are similar, we know that the corresponding angles are equal. In particular, we have
∠F = ∠C
and
∠E = ∠B
Since ∠B is a right angle, we have ∠E = 90°.
Using the fact that the sides of triangle DEF are 1/3 the length of the corresponding sides of triangle ABC, we have
DE = 1/3 AB = 4
and
EF = 1/3 BC = 16/3
Now we can use the Pythagorean theorem to find DF:
DF^2 = DE^2 + EF^2 = 4^2 + (16/3)^2 = 256/9
So DF = √(256/9) = 16/3.
Finally, we can use the sine function to find sin(F):
sin(F) = DF/EF = (16/3) / (16/3) = 1.
Therefore, sin(F) = 1.
Thank you.
You're welcome!
The final answer is
3/5
or 0.6.